MEDIAN
DEFINITION
Median simply refers to a value or observation in the middle of a set of values.
In other words, it is a value that separates the upper and lower halves of a set of values or even a distribution. Median is a commonly used measure of central tendency.
CALCULATOR

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Median:
FORMULA AND DERIVATION

Let us consider a simple set of observations $$1,2,3$$ . It is very clear that the value in the middle is 2, as indicated below

1 ② 3.

Therefore, the median of the above set of data is 2.

Let us consider another set of observations $$1,2,3,4$$ . In this case there is no value that lies exactly in the center. Instead, there are 2 value in the middle.

1 ② ③ 4.

Since 2 and 3 are in the middle, we consider the average of these two terms to be the median. Therefore, the median is $$\frac{2+3}{2}=2.5$$

So, in general, for $$n$$ observations, the median is the $${\left(\frac{n+1}{2}\right)}^{th}$$ term. Before finding the median, it is necessary that the observations be arranged in either ascending or descending order.

In case each $$x_i$$ is repeated $$f_i$$ times, i.e. if $$x_i$$has a frequency of $$f_i$$ as follows:-

Observations $$\left(x_i\right)$$ Frequencies$$\left(f_i\right)$$
$$x_1$$ $$f_1$$
$$x_2$$ $$f_2$$
... ...
... ...
$$x_n$$ $$f_n$$
$$\sum{f}=N$$

then, the total number of observations can be obtained by summing $$f_i$$, i.e. total number of observation $$=N=\sum{f}$$

In this case, we can find the $${\left(\frac{n+1}{2}\right)}^{th}$$ observation by summing up the frequencies, until we reach the $${\left(\frac{n+1}{2}\right)}^{th}$$ term.

This can be done by finding the cumulative frequencies as follows:

Observations $$\left(x_i\right)$$ Frequencies$$\left(f_i\right)$$ Cumulative Frequencies
$$x_1$$ $$f_1$$ $$f_1$$
$$x_2$$ $$f_2$$ $$f_1+f_2$$
$$x_3$$ $$f_3$$ $$f_1+f_2+f_3$$
... ... ...
... ... ...
$$x_n$$ $$f_n$$ $$f_1+f_2+f_3\dots \dots f_n$$
$$\sum{f}=N$$

We must now find that cumulative frequency value which is just greater than $$\frac{n+1}{2}$$ . The corresponding $$x$$ value to that cumulative frequency is the median.

EXAMPLES
Example 1

Consider the following observations:

$$8,6,5,9,14,12,10$$

To find the median, the observations must be arranged in ascending or descending order. In ascending order, the observations are:

$$5,6,8,9,10,12,14$$

number of observations $$=n=7$$

Therefore, median is the $${\left(\frac{7+1}{2}\right)}^{th}$$ value, that is the 4th value. From the list of observations, it can be seen that the 4th value is 9.

Therefore, 9 is the median.

Example 2

Consider the following frequency distribution

Observations $$(x_i)$$ Frequencies$$\left(f_i\right)$$
$$2$$ $$4$$
$$3$$ $$6$$
$$4$$ $$4$$
$$5$$ $$7$$
$$6$$ $$3$$
$$7$$ $$2$$
$$8$$ $$3$$
$$N=29$$

Since, $$n=29$$ the median is the $${\left(\frac{29+1}{2}\right)}^{th}$$ value i.e. $$15th\ value$$.

From the frequency distribution it can be observed that the first four values are 2, the next 6 value are 3, and the next 4 values are 4. So far, we have counted $$4+6+4=14$$ values. But the median is the $$15th\ value$$. Since 4 is the $$14th\ value$$, the $$15th\ value$$ is 5. And therefore, 5 is the median.

This can be easily identified by calculating the cumulative frequencies, as follows:

Observations $$\left(x_i\right)$$ Frequencies$$\left(f_i\right)$$ Cumulative Frequencies
$$2$$ $$4$$ $$4$$
$$3$$ $$6$$ $$10$$
$$4$$ $$4$$ $$14$$
$$5$$ $$7$$ $$21$$
$$6$$ $$3$$ $$24$$
$$7$$ $$2$$ $$26$$
$$8$$ $$3$$ $$29$$
$$N=29$$

From the cumulative frequency distribution, it can be observed that the $$14th\ value$$ is 4 and the $$15th\ value$$ must be the next observation, i.e. 5.

Therefore, we must observe the cumulative frequency which is just higher than 15, which is 21 in this case. The median is the observation corresponding to this cumulative frequency.

RELATED TOPICS
Mean
Mode